Maximum Subsequence Score

Maximum Subsequence Score

1. What is this problem about?

The Maximum Subsequence Score problem involves two arrays, nums1 and nums2, of the same length, and an integer k. You need to choose k indices to form a subsequence. Your score is calculated as the sum of elements from nums1 at these indices multiplied by the minimum of the elements from nums2 at these same indices. The goal is to maximize this score. It’s a balancing act: you want high values in nums1 to increase the sum, but you are limited by the smallest value in nums2 across your chosen indices.

2. Why is this asked in interviews?

This "Maximum Subsequence Score interview question" is popular at Microsoft, Amazon, and Google. It is a brilliant test of a candidate's ability to combine sorting with a priority queue (heap). It evaluates whether you can recognize that by sorting based on one array (the multiplier), you can simplify the problem of finding the best set of elements for the other array (the sum). It’s a classic "greedy with a heap" problem.

3. Algorithmic pattern used

The "Array, Sorting, Heap (Priority Queue), Greedy interview pattern" is the way to go. First, pair the elements from nums1 and nums2 and sort them in descending order based on nums2. This way, as you iterate through the sorted pairs, the current nums2 value is the minimum possible value for any subsequence ending at this point. You then use a Min-Heap to keep track of the k largest elements from nums1 seen so far to maximize the sum part of the score.

4. Example explanation

nums1 = [1, 3, 3, 2], nums2 = [2, 1, 3, 4], k = 3.

• Pairs (n1, n2): (1,2), (3,1), (3,3), (2,4).
• Sort by n2 descending: (2,4), (3,3), (1,2), (3,1).
• Step 1: Pick (2,4). Heap: [2], Sum: 2. Not enough for k=3.
• Step 2: Pick (3,3). Heap: [2, 3], Sum: 5. Not enough for k=3.
• Step 3: Pick (1,2). Heap: [1, 2, 3], Sum: 6. Multiplier is 2 (min of 4,3,2). Score = 6 * 2 = 12.
• Step 4: Pick (3,1). Heap: [2, 3, 3] (dropped 1), Sum: 8. Multiplier is 1. Score = 8 * 1 = 8. Max score is 12.

5. Common mistakes candidates make

In the "Maximum Subsequence Score coding problem," candidates often try to sort by nums1 or use dynamic programming, which results in inefficient solutions. Another mistake is using a Max-Heap instead of a Min-Heap; you need a Min-Heap to quickly identify and remove the smallest element in your current set of k values to keep the total sum as high as possible. Some also forget that the multiplier is the current element's nums2 value because we sorted in descending order.

6. Interview preparation tip

Practice problems that require "sorting by one criteria and using a heap for another." This is a very common pattern for optimization problems involving two related arrays. Get comfortable with Python’s heapq or Java’s PriorityQueue. When explaining your solution, emphasize why sorting by nums2 is the "unlock" that allows you to treat the minimum value as a fixed variable while you greedily pick the best nums1 values.